%%
%% lecture22.tex
%% 
%% Made by alex
%% Login   <alex@tomato>
%% 
%% Started on  Thu Jan  5 08:30:44 2012 alex
%% Last update Thu Jan  5 08:30:44 2012 alex
%%





\exercises
\begin{xca}
Let us assume that $M$ is a closed connected orientable
three-dimensional manifold. Prove that all homology and
cohomology groups can be expressed in terms of its fundamental
group $\pi_1(M)$. Calculate the homology and cohomology with
integer coefficients if $\pi_1(M)=\ZZ_m$.
\end{xca}
\begin{xca}
An $n$-dimensional compact manifold with boundary is
contractible. Prove that its boundary is a homology sphere
(i.e. it has the same homology as $(n-1)$-dimensional sphere).
\end{xca}
\begin{xca}
Let us denote by $T$ a graph consisting of all edges of a
tetrahedron. Calculate the homology and cohomology of the
complement of $T$ in $\RR^3$.
\end{xca}
